Fermat Quotient

The Fermat quotient for a number $a$ and a Prime base $p$ is defined as

$$q_p(a)\equiv {a^{p-1}-1\over p}.$$ (1)

If $p\notdiv ab$, then

$$q_p(ab) = q_p(a)+q_p(b)$$ (2)

$$q_p(p\pm 1) = \mp 1$$ (3)

$$q_p(2) = {1\over p} \left({1-{1\over 2}+{1\over 3}-{1\over 4}+\ldots-{1\over p-1}}\right),$$ (4)

all (mod $p$). The quantity $q_p(2)=(2^{p-1}-1)/p$ is known to be Square for only two Primes: the so-called Wieferich Primes 1093 and 3511 (Lehmer 1981, Crandall 1986).

See also Wieferich Prime

References

Crandall, R. Projects in Scientific Computation. New York: Springer-Verlag, 1986.

Lehmer, D. H. ``On Fermat's Quotient, Base Two.'' Math. Comput. 36, 289-290, 1981.